최적화를 위한 테스트 함수

${\displaystyle f(\mathbf {x} )=An+\sum _{i=1}^{n}\왼쪽[x_{i}^{2}-A\cos(2\pi x_{i}\ri}\right]}}}$

${\displaystyle {\text{where:}A=10}$

${\displaystyle f(x,y)=-20\exp \좌측[-0.2{\sqrt {0.5\좌측(x^{2}+y^{2}\우측)}}}$

${\displaystyle -\exp \좌측[0.5\좌측(\cos 2\pi x+\cos 2\pi y\우측)\우측]+e+20}$

${\displaystyle f(x,y)=\좌측(1.5-x+xy\오른쪽)^{2}+\좌측(2.25-x+xy^{2}\우측)^{2}}$

${\displaystyle +\왼쪽(2.625-x+xy^{3}\오른쪽)^{2}}$

${\displaystyle f(x,y)=\좌측[1+\좌측(x+y+1\우측)^{2}\좌측(19-14x+3x^{2}-14y+6xy^{2}\우측)}}$

${\displaystyle \left[30+\left(2x-3y\오른쪽)^{2}\{2}\reft (18-32x+12x^{2}+48y-36xy+27y^{2}\right]}$

${\displaystyle f(x,y)=\sin ^{2}3\pi x+\좌측(x-1\우측)^{2}\좌측(1+\sin ^{2}3\pi y\우측)}$

${\displaystyle +\좌측(y-1\우측)^{2}\좌측(1+\sin ^{2}2\pi y\우측)}$

${\displaystyle f}$( ${\displaystyle x}$, ${\displaystyle y}$)${\displaystyle }$=${\displaystyle }$( ${\displaystyle 1}$- ${\displaystyle x{}^{}}$) 2${\displaystyle {}^{}}$+ ${\displaystyle 100}$ ${\displaystyle (y}$- ${\displaystyle {}^{}}$ ${\displaystyle {2\mathrm{}}^{}{}^{}}$) ${\displaystyle {2\mathrm{}}^{}}$ ${\$

대상: (${\displaystyle \mathrm{x}}$- 1${\displaystyle {}^{}}$) 3 -${\displaystyle {}^{}}$y${\displaystyle }$+ ${\displaystyle 1}$ ${\displaystyle \le }$ 0 ${\displaystyle 및}$ ${\displaystyle \text{x}}$+ y${\displaystyle }$- ${\displaystyle 2}$ ${\displaystyle \le }$ 0 ${\displaystyle (x-1)^{3}-y+1\leq$ 0${\text{$및$}x+y-2\leq 0}$

${\displaystyle f}$( ${\displaystyle x}$, ${\displaystyle y}$)${\displaystyle }$=${\displaystyle }$( ${\displaystyle 1}$- ${\displaystyle x{}^{}}$) 2${\displaystyle {}^{}}$+ ${\displaystyle 100}$ ${\displaystyle (y}$- ${\displaystyle {}^{}}$ ${\displaystyle {2\mathrm{}}^{}{}^{}}$) ${\displaystyle {2\mathrm{}}^{}}$ ${\$

대상: $x{\displaystyle {}^{}}$ ${\displaystyle {2\mathrm{}}^{}}$+ ${\displaystyle {}^{}}$ ${\displaystyle {2\mathrm{}}^{}}$ ${\displaystyle {}^{}}$ ${\displaystyle 2}$ ${\displaystyle$ x$^{2}+y^{2$}$}\leq 2}$

${\displaystyle f(x,y)=\sin(y)e^{\left[(1-\cos x)^{2}\right]}+\cos(x)e^{\left[(1-\sin y)^{2}\right]}+(x-y)^{2}}$,

: (x+ ) +( + ) < ${\displaystyle (x+5)^{2}+(y+5)^{2$$}<25}$

${\displaystyle f}$( x${\displaystyle }$, ${\displaystyle y}$)${\displaystyle }$=${\displaystyle }$-${\displaystyle }$[ ${\displaystyle \cos }$ ${\displaystyle }$(${\displaystyle }$( x${\displaystyle \mathrm{}}$- 0.${\displaystyle 1}$) ${\displaystyle y}$ )${\displaystyle {}^{}}$ ${\displaystyle {2\mathrm{}}^{}}$- ${\displaystyle x}$ ${\displaystyle \sin }$ ${\displaystyle }$( ${\displaystyle 3}$ ${\displaystyle x}$+ ${\displaystyle y}$) ${\displaystyle f(x,y)=-[\cos(x-0.1)y)]^{2}-x\sin(3x+y$

대상: x + <[ 2 t - 2 2 t - 2 cos ⁡ - 4 t - + [ ] ] [\$displaysty$ x$^{2}+y$^{2]{$2$$.$$}}<왼쪽[2\cos t-{\frac {1}{1$}:{2$}}\cos 2t-{\prac {1}{$1}{$4}}\cos 3t-{1}{8}\cos 4t\right]^{2$$}+[2\sin t]^{2}}:$ t = Atan2(x,y)

${\displaystyle f(}$ ${\displaystyle x}$, y${\displaystyle }$)${\displaystyle }$= 4 ${\displaystyle {}^{}2{}^{}}$- 2.1 x ${\displaystyle {}^{}}$+ ${\displaystyle 1\frac{\mathrm{}}{}}$ 3 ${\displaystyle x{}^{}}$ ${\displaystyle {6\mathrm{}}^{}}$+ ${\displaystyle 4}$ ${\displaystyle {}^{}{2\mathrm{}}^{}}$+ ${\displaystyle 4}$ ${\displaystyle \mathrm{y}}$ ${\displaystyle {4\mathrm{}}^{}}$ ${\$1}{1$}{3}}}x^{6}+xy-4y^{4$4$},}},}}}},}}}}}}},},$

대상: -${\displaystyle \sin }$ ${\displaystyle }$(${\displaystyle 4}$ ${\displaystyle \pi }$ x${\displaystyle }$)${\displaystyle }$+ ${\displaystyle 2}$ ${\displaystyle {\sin }^{}}$ ${\displaystyle {2\mathrm{}}^{}}$ ${\displaystyle {}^{}}$( 2 ${\displaystyle \pi }$ y${\displaystyle }$)${\displaystyle \mathrm{}\le }$ ${\displaystyle 1}$.5 ${\displaystyle -\sin (4\pi$ x$)+2\sin ^{2}(2\pi$ y$)\leq$1.5$}$

${\displaystyle f}$( x${\displaystyle }$, ${\displaystyle y}$)${\displaystyle }$=${\displaystyle 0}$.1 ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle f(x,y)=0.1xy$

대상: ${\displaystyle {x\mathrm{}}^{}}$ 2${\displaystyle {}^{}}$+ y ${\displaystyle {2\mathrm{}}^{}}$ ${\displaystyle {}^{}{}^{}}$[ ${\displaystyle {r{}_{}}^{}}$ ${\displaystyle {T_{}}^{}}$+ ${\displaystyle {r_{}}^{}}$ S ${\displaystyle {\cos }^{}}$ ${\displaystyle {}^{}}$(${\displaystyle n^{}}$ ${\displaystyle {\arctan }^{}}$ ${\displaystyle {}^{}}$ x ${\displaystyle {\frac{y}{}}^{}}$) ${\displaystyle {]}^{}}$ ${\displaystyle {\mathrm{}}^{}}$ ${\$ x$^{2}+y^{2$}$}\leq \left[r_{$$T}+r_{S}\cos \left(n\arctan {\frac {{x}{y}\right)\2}}:$ ${\displaystyle 여기}$서${\displaystyle \text{:}}$ ${\displaystyle r_{}\mathrm{T}}$ =${\displaystyle {}_{}}$, ${\displaystyle r_{}}$S = 0.${\displaystyle 2}$ ${\displaystyle \mathrm{및}n}$ = 8$${\$displaystyle${\$text{where: }r_${}{}{}$r_$${{}}}}}}}}$$T}=1,r_{S}=0.2{\text{ 및 }}n=8}$

${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=\left[1+\left(A_{1}-B_{1}\left(x,y\right)\right)^{2}+\left(A_{2}-B_{2}\left(x,y\right)\right)^{2}\right]\\f_{2}\left(x,y\right)=\left(x+3\right)^{2}+\left(y+1\right)^{2}\\\end{cases}}}$

${\displaystyle {\text{where}={\reason{case}A_{1}=0.5\sin \왼쪽(1\오른쪽)-2\cos \left(1\오른쪽)+\sin \reft(2\오른쪽)-1.5\cos \reft(2\ 오른쪽)\\cos \refted(2\오른쪽)\\\\\cos \coses \reft;A_{2}=1.5\sin \왼쪽(1\오른쪽)-\cos \왼쪽(1\오른쪽)+2\sin \왼쪽(2\오른쪽)-0.5\cos \왼쪽(2\오른쪽)\\B_{1}\좌측(x,y\오른쪽)=0.5\sin \좌측(x\오른쪽)-2\cos \좌측(x\오른쪽)\sin \좌측(y\오른쪽)+\sin \cos \1.5\cos \좌측(y\오른쪽)\\B_{2}\좌(x,y\오른쪽)=1.5\sin \좌(x\오른쪽)-\cos \좌(x\오른쪽)+2\sin \좌(y\오른쪽)-0.5\cos \left(y\오른쪽)\end{case}}$